direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C42, C23⋊3C42, C84⋊14C22, C42.58C23, C4⋊(C2×C42), C28⋊9(C2×C6), (C2×C4)⋊2C42, (C2×C84)⋊14C2, (C2×C28)⋊14C6, (C2×C12)⋊6C14, C12⋊4(C2×C14), C22⋊2(C2×C42), (C22×C42)⋊1C2, (C2×C42)⋊8C22, (C22×C6)⋊3C14, (C22×C14)⋊9C6, C2.1(C22×C42), C6.11(C22×C14), C14.25(C22×C6), (C2×C6)⋊2(C2×C14), (C2×C14)⋊11(C2×C6), SmallGroup(336,205)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C42
G = < a,b,c | a42=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 140 in 108 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C7, C2×C4, D4, C23, C12, C2×C6, C2×C6, C2×C6, C14, C14, C14, C2×D4, C21, C2×C12, C3×D4, C22×C6, C28, C2×C14, C2×C14, C2×C14, C42, C42, C42, C6×D4, C2×C28, C7×D4, C22×C14, C84, C2×C42, C2×C42, C2×C42, D4×C14, C2×C84, D4×C21, C22×C42, D4×C42
Quotients: C1, C2, C3, C22, C6, C7, D4, C23, C2×C6, C14, C2×D4, C21, C3×D4, C22×C6, C2×C14, C42, C6×D4, C7×D4, C22×C14, C2×C42, D4×C14, D4×C21, C22×C42, D4×C42
►On 168 points
Generators in S168
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 84 102 154)(2 43 103 155)(3 44 104 156)(4 45 105 157)(5 46 106 158)(6 47 107 159)(7 48 108 160)(8 49 109 161)(9 50 110 162)(10 51 111 163)(11 52 112 164)(12 53 113 165)(13 54 114 166)(14 55 115 167)(15 56 116 168)(16 57 117 127)(17 58 118 128)(18 59 119 129)(19 60 120 130)(20 61 121 131)(21 62 122 132)(22 63 123 133)(23 64 124 134)(24 65 125 135)(25 66 126 136)(26 67 85 137)(27 68 86 138)(28 69 87 139)(29 70 88 140)(30 71 89 141)(31 72 90 142)(32 73 91 143)(33 74 92 144)(34 75 93 145)(35 76 94 146)(36 77 95 147)(37 78 96 148)(38 79 97 149)(39 80 98 150)(40 81 99 151)(41 82 100 152)(42 83 101 153)
(1 154)(2 155)(3 156)(4 157)(5 158)(6 159)(7 160)(8 161)(9 162)(10 163)(11 164)(12 165)(13 166)(14 167)(15 168)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 133)(23 134)(24 135)(25 136)(26 137)(27 138)(28 139)(29 140)(30 141)(31 142)(32 143)(33 144)(34 145)(35 146)(36 147)(37 148)(38 149)(39 150)(40 151)(41 152)(42 153)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)
G:=sub<Sym(168)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,84,102,154)(2,43,103,155)(3,44,104,156)(4,45,105,157)(5,46,106,158)(6,47,107,159)(7,48,108,160)(8,49,109,161)(9,50,110,162)(10,51,111,163)(11,52,112,164)(12,53,113,165)(13,54,114,166)(14,55,115,167)(15,56,116,168)(16,57,117,127)(17,58,118,128)(18,59,119,129)(19,60,120,130)(20,61,121,131)(21,62,122,132)(22,63,123,133)(23,64,124,134)(24,65,125,135)(25,66,126,136)(26,67,85,137)(27,68,86,138)(28,69,87,139)(29,70,88,140)(30,71,89,141)(31,72,90,142)(32,73,91,143)(33,74,92,144)(34,75,93,145)(35,76,94,146)(36,77,95,147)(37,78,96,148)(38,79,97,149)(39,80,98,150)(40,81,99,151)(41,82,100,152)(42,83,101,153), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,161)(9,162)(10,163)(11,164)(12,165)(13,166)(14,167)(15,168)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,84,102,154)(2,43,103,155)(3,44,104,156)(4,45,105,157)(5,46,106,158)(6,47,107,159)(7,48,108,160)(8,49,109,161)(9,50,110,162)(10,51,111,163)(11,52,112,164)(12,53,113,165)(13,54,114,166)(14,55,115,167)(15,56,116,168)(16,57,117,127)(17,58,118,128)(18,59,119,129)(19,60,120,130)(20,61,121,131)(21,62,122,132)(22,63,123,133)(23,64,124,134)(24,65,125,135)(25,66,126,136)(26,67,85,137)(27,68,86,138)(28,69,87,139)(29,70,88,140)(30,71,89,141)(31,72,90,142)(32,73,91,143)(33,74,92,144)(34,75,93,145)(35,76,94,146)(36,77,95,147)(37,78,96,148)(38,79,97,149)(39,80,98,150)(40,81,99,151)(41,82,100,152)(42,83,101,153), (1,154)(2,155)(3,156)(4,157)(5,158)(6,159)(7,160)(8,161)(9,162)(10,163)(11,164)(12,165)(13,166)(14,167)(15,168)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90)(73,91)(74,92)(75,93)(76,94)(77,95)(78,96)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,84,102,154),(2,43,103,155),(3,44,104,156),(4,45,105,157),(5,46,106,158),(6,47,107,159),(7,48,108,160),(8,49,109,161),(9,50,110,162),(10,51,111,163),(11,52,112,164),(12,53,113,165),(13,54,114,166),(14,55,115,167),(15,56,116,168),(16,57,117,127),(17,58,118,128),(18,59,119,129),(19,60,120,130),(20,61,121,131),(21,62,122,132),(22,63,123,133),(23,64,124,134),(24,65,125,135),(25,66,126,136),(26,67,85,137),(27,68,86,138),(28,69,87,139),(29,70,88,140),(30,71,89,141),(31,72,90,142),(32,73,91,143),(33,74,92,144),(34,75,93,145),(35,76,94,146),(36,77,95,147),(37,78,96,148),(38,79,97,149),(39,80,98,150),(40,81,99,151),(41,82,100,152),(42,83,101,153)], [(1,154),(2,155),(3,156),(4,157),(5,158),(6,159),(7,160),(8,161),(9,162),(10,163),(11,164),(12,165),(13,166),(14,167),(15,168),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,133),(23,134),(24,135),(25,136),(26,137),(27,138),(28,139),(29,140),(30,141),(31,142),(32,143),(33,144),(34,145),(35,146),(36,147),(37,148),(38,149),(39,150),(40,151),(41,152),(42,153),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90),(73,91),(74,92),(75,93),(76,94),(77,95),(78,96),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6N | 7A | ··· | 7F | 12A | 12B | 12C | 12D | 14A | ··· | 14R | 14S | ··· | 14AP | 21A | ··· | 21L | 28A | ··· | 28L | 42A | ··· | 42AJ | 42AK | ··· | 42CF | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 7 | ··· | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C7 | C14 | C14 | C14 | C21 | C42 | C42 | C42 | D4 | C3×D4 | C7×D4 | D4×C21 |
| kernel | D4×C42 | C2×C84 | D4×C21 | C22×C42 | D4×C14 | C2×C28 | C7×D4 | C22×C14 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C42 | C14 | C6 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 6 | 6 | 24 | 12 | 12 | 12 | 48 | 24 | 2 | 4 | 12 | 24 |
Matrix representation of D4×C42 ►in GL3(𝔽337) generated by
| 336 | 0 | 0 |
| 0 | 333 | 0 |
| 0 | 0 | 333 |
| 336 | 0 | 0 |
| 0 | 336 | 335 |
| 0 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 336 | 335 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(337))| [336,0,0,0,333,0,0,0,333],[336,0,0,0,336,1,0,335,1],[1,0,0,0,336,0,0,335,1] >;
D4×C42 in GAP, Magma, Sage, TeX
D_4\times C_{42} % in TeX
G:=Group("D4xC42"); // GroupNames label
G:=SmallGroup(336,205);
// by ID
G=gap.SmallGroup(336,205);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-7,-2,2041]);
// Polycyclic
G:=Group<a,b,c|a^42=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations